Problem: $\dfrac{ t - 8u }{ -6 } = \dfrac{ 10t - 8v }{ -4 }$ Solve for $t$.
Answer: Multiply both sides by the left denominator. $\dfrac{ t - 8u }{ -{6} } = \dfrac{ 10t - 8v }{ -4 }$ $-{6} \cdot \dfrac{ t - 8u }{ -{6} } = -{6} \cdot \dfrac{ 10t - 8v }{ -4 }$ $t - 8u = -{6} \cdot \dfrac { 10t - 8v }{ -4 }$ Multiply both sides by the right denominator. $t - 8u = -6 \cdot \dfrac{ 10t - 8v }{ -{4} }$ $-{4} \cdot \left( t - 8u \right) = -{4} \cdot -6 \cdot \dfrac{ 10t - 8v }{ -{4} }$ $-{4} \cdot \left( t - 8u \right) = -6 \cdot \left( 10t - 8v \right)$ Distribute both sides $-{4} \cdot \left( t - 8u \right) = -{6} \cdot \left( 10t - 8v \right)$ $-{4}t + {32}u = -{60}t + {48}v$ Combine $t$ terms on the left. $-{4t} + 32u = -{60t} + 48v$ ${56t} + 32u = 48v$ Move the $u$ term to the right. $56t + {32u} = 48v$ $56t = 48v - {32u}$ Isolate $t$ by dividing both sides by its coefficient. ${56}t = 48v - 32u$ $t = \dfrac{ 48v - 32u }{ {56} }$ All of these terms are divisible by $8$ $t = \dfrac{ {6}v - {4}u }{ {7} }$